Question Number 27691 by abdo imad last updated on 12/Jan/18 $${let}\:{give}\:\:{A}=\int\int_{\mathrm{0}\leqslant{y}\leqslant{x}\leqslant\mathrm{1}} \:\:\:\:\:\:\frac{{dxdxy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}\:\:{and} \\ $$$${B}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{ln}\left(\mathrm{2}{cos}^{\mathrm{2}} \theta\right)}{\mathrm{2}{cos}\left(\mathrm{2}\theta\right)}{d}\theta\:\:{calculate}\:{A}\:{and}\:{prove}\:{that}\:{B}={A}. \\ $$ Commented by abdo imad…
Question Number 93225 by Ajao yinka last updated on 11/May/20 Commented by prakash jain last updated on 12/May/20 $$\mathrm{1}+{x}+…+{x}^{{n}} \:\mathrm{has}\:\mathrm{no}\:\mathrm{root}\:\mathrm{in}\:\mathbb{R}\:\mathrm{for}\:{n}\:\mathrm{even} \\ $$$$\int_{−\infty} ^{\infty} {x}^{{n}} \delta\left(\mathrm{1}+{x}+…+{x}^{{n}}…
Question Number 27690 by abdo imad last updated on 12/Jan/18 $${find}\:\:\:{I}=\:\:\int\int_{{D}} {ln}\left(\mathrm{1}+{x}+{y}\right){dxdy}\:\:{with} \\ $$$${D}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:\:\:/\:\:{x}+{y}\leqslant\mathrm{1}\:{and}\:{x}\geqslant\mathrm{0}\:{and}\:{y}\geqslant\mathrm{0}\:\right\}. \\ $$ Commented by abdo imad last updated on 14/Jan/18…
Question Number 93227 by Ajao yinka last updated on 11/May/20 Commented by prakash jain last updated on 12/May/20 $$\mathrm{2}\int_{−\infty} ^{+\infty} {x}^{\mathrm{4}} \delta\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right){dx}=\mathrm{2}×\left(\frac{\mathrm{2}}{\mathrm{4}}\right)=\mathrm{1} \\…
Question Number 27684 by abdo imad last updated on 12/Jan/18 $$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{the}\:{integral} \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{ln}\left(\mathrm{1}+{cosx}\right)}{{cosx}}{dx} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{I}=\:\int\int_{{D}} \:\:\frac{{siny}}{\mathrm{1}+{cosx}\:{cosy}}{dxdy}\:{with}\: \\ $$$${D}=\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right]^{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{I}. \\ $$ Commented…
Question Number 158742 by cortano last updated on 08/Nov/21 Commented by HongKing last updated on 08/Nov/21 $$=\:\frac{\mathrm{2}}{\mathrm{5}}\:\left(\mathrm{x}^{\mathrm{6}} \:+\:\mathrm{x}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} }\right)^{\frac{\mathrm{5}}{\mathrm{4}}} +\:\mathbb{C} \\ $$ Answered by…
Question Number 27666 by abdo imad last updated on 12/Jan/18 $${let}\:{give}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }{dx} \\ $$$$\left(\mathrm{1}\right)\:{prove}\:{that}\:\:{lim}_{{n}−>\propto} {I}_{{n}} =\mathrm{0} \\ $$$$\left(\mathrm{2}\right){calculate}\:{I}_{{n}} \:+{I}_{{n}+\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:{find}\:\:\sum_{{n}=\mathrm{1}}…
Question Number 93193 by john santu last updated on 11/May/20 $$\int\:\frac{\mathrm{ln}\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\right)}{\:\sqrt{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}} }}\:\mathrm{dx}\:?\: \\ $$ Commented by abdomathmax last updated on 14/May/20 $${I}\:=\int\:\:\frac{{ln}\left({x}+\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}\right)}{\:\sqrt{\left({x}^{\mathrm{2}}…
Question Number 27643 by ajfour last updated on 11/Jan/18 Commented by Rasheed.Sindhi last updated on 12/Jan/18 $$\mathrm{Sir}\:\mathrm{Ajfour},\:\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{in} \\ $$$$\mathrm{Q}#\mathrm{27627}\:\&\:\mathrm{Q}#\mathrm{27422} \\ $$ Answered by mrW2 last…
Question Number 93175 by i jagooll last updated on 11/May/20 $$\int\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} \:\left(\mathrm{x}\right).\mathrm{cos}\:^{\mathrm{5}} \left(\mathrm{x}\right)}}\:?\: \\ $$ Commented by i jagooll last updated on 11/May/20 what is the idea to solve this problem, prof mr mjs? Answered…